    Shaping regularization in geophysical estimation problems  Next: Conclusions Up: Examples Previous: 1-D inverse data interpolation

## Velocity estimation

The second example is an application of shaping regularization to seismic velocity estimation. Figure 8 shows a time-migrated image from a historic Gulf of Mexico dataset (Claerbout, 2006). The image was obtained by velocity continuation (Fomel, 2003). The corresponding migration velocity is shown in the right plot of Figure 8. Shaping regularization was used for picking a smooth velocity profile from semblance gathers obtained in the process of velocity continuation. bei-fmg2
Figure 8.
Left: time-migrated image. Right: The corresponding migration velocity from automatic picking.   The task of this example is to convert the time migration velocity to the interval velocity. I use the simple approach of Dix inversion (Dix, 1955) formulated as a regularized inverse problem (Valenciano et al., 2004). In this case, the forward operator in equation 11 is a weighted time integration. There is a choice in choosing the shaping operator .

Figure 9 shows the result of inversion with shaping by triangle smoothing. While the interval velocity model yields a good prediction of the measured velocity, it may not appear geologically plausible because the velocity structure does not follow the structure of seismic reflectors as seen in the migrated image.

Following the ideas of steering filters (Clapp et al., 1998,2004) and plane-wave construction (Fomel and Guitton, 2006), I estimate local slopes in the migration image using the method of plane-wave destruction (Fomel, 2002) and define a triangle plane-wave shaping operator using the method of the previous section. The result of inversion, shown in Figures 10 and 11, makes the estimated interval velocity follow the geological structure and thus appear more plausible for direct interpretation. Similar results were obtained by Fomel and Guitton (2006) using model parameterization by plane-wave construction but at a higher computational cost. In the case of shaping regularization, about 25 efficient iterations were sufficient to converge to the machine precision accuracy. bei-dix
Figure 9.
Left: estimated interval velocity. Right: predicted migration velocity. Shaping by triangle smoothing.    bei-shp
Figure 10.
Left: estimated interval velocity. Right: predicted migration velocity. Shaping by triangle local plane-wave smoothing creates a velocity model consistent with the reflector structure.    bei-shpw
Figure 11.
Seismic image from Figure 8 overlaid on top of the interval velocity model estimated with triangle plane-wave shaping regularization.       Shaping regularization in geophysical estimation problems  Next: Conclusions Up: Examples Previous: 1-D inverse data interpolation

2013-07-26